Active contours, or snakes [10] have found many applications
in image analysis and computer vision. A snake is a parametric contour
, in the image plane
(x,y). The snake evolves under internal and external ``forces'',
usually described by energy terms 
and 
. The
internal forces keep the snake smooth and the external forces couple
the snake to the image I(x,y), attracting the snake to features of
interest (for segmentation applications, these features would be object
boundaries). The internal forces usually take the form
The 
term controls the ``tension'' of the snake while the
term controls its ``rigidity''. The external forces take a
similar form:
where P(x,y) is a scalar potential function designed so that local
minima of the function coincide with image features of interest. For
instance, if we are trying to detect boundaries which are
characterised by high intensity gradients, then a suitable potential
would be 
, where 
is a 2D Gaussian kernel with standard
deviation 
pixels.
Starting from an initial position, the snake evolves under the
influence of the combined field 
. In practical
implementations, the snake is split into piecewise linear
segments and local forces are calculated at each vertex. The motion of
the snake is computed using gradient descent [10] or
Lagrangian dynamics [11]. For segmentation applications,
the idea is that the equilibrium position of the snake provides the
desired segmentation. If I(x,y) changes, the snake can also be used
to track a boundary over time, providing motion as well as
shape information [3].
The main reason for using snakes in segmentation is that they allow the incorporation of geometric constraints. In particular, they exploit (in fact they rely on) prior boundary estimates obtained manually or from the previous image in a sequence. Such an approach sensibly combines the strengths of human operators (initial boundary recognition) and computer algorithms (accurate boundary delineation). Snakes have proved very successful for medical image segmentation [12]. In particular, snakes have been shown to give state-of-the-art performance for segmenting intra-vascular ultrasound images [4].
An alternative way to enforce the smoothness constraint is to use
cubic B-spline snakes [6]. A cubic B-spline is specified
by m+1 control points 
and
comprises m-2 cubic polynomial curve segments 
. The joining points between each curve segment are
known as knots. The equation of each segment is \
for 
and 
. For notational convenience we
can re-parameterise the spline over a single parameter s:
where 
are the spline basis functions and 
. The
representation 
is more compact than the
long list of vertices in piecewise linear formulations. Moreover, the
internal forces are no longer required: the smoothness of the snake is
implicit in the B-spline formulation. B-splines may be open or closed
as required, and are defined with continuity properties at each
knot. The flexibility of the curve increases as more control points
are added: each additional control point allows one more
inflection. It is also possible to use multiple knots to reduce the
continuity at knots [14].
However, what makes B-splines especially useful for segmentation is
that they exhibit local control: modifying the position of one control
point causes only a small part of the curve to change. This leads to
an effective and simple algorithm for computing the motion of the
snake under the influence of image forces -- see
Figure 1. Local forces are computed at n sample
points 
distributed evenly around the
spline. At each sample point, a one-dimensional search is performed
normal to the spline for a local minimum 
of P(x,y): we
shall refer to the points 
as
target points. A force is applied at each sample point proportional
to the distance between the sample and the target points, ie:
The positions of the control points are then updated to minimize
using linear least squares [6]. The
one-dimensional search and re-estimation of the control points is
repeated until the snake attains equilibrium.
Figure 1: B-spline snakes. The snake evolves under a set of
``forces'' applied at a set of sample points. Each force is
proportional to the distance between the sample point and the target
point. The target point is found by a one-dimensional search normal to
the spline for an image feature of interest. At equlibrium the snake
provides a compact representation of the boundary.
The main drawback of both piecewise linear and B-spline snakes is the
difficulty in specifying a-priori the correct smoothness
parameters (ie. setting and , or specifying the
correct number of control points). Techniques have been proposed to
automatically introduce the ``best'' number of control
points [5], though such techniques tend to be
computationally expensive. Here we propose an extremely
straightforward, novel work-around. We use a B-spline formulation with
a fairly small number of control points. While this tends to give
overly smoothed contours, we base our segmentation not on the B-spline
but on a piecewise linear contour with vertices . The target points provide a good segmentation,
while the B-spline formulation constrains the search for the target
points and therefore stops the snake becoming too ``bent'' and
intersecting itself.
